Roderick B.

Astillero

Probability theory

Bayesian Probability

BAYESIAN PROBABILITY
THE TOTAL PROBABILITY The total probability or rule of elimination provides us with a convenience method for solving the probability of any event that is common between two or more mutually exclusive events. Theorem 5.1. Theorem of Total Probability If the events B1, B2, …, Bk constitute a partition of the sample space S such that P(Bi) ≠ 0 for i = 1, 2, 3,…,k, then for any event A of S,
k i=1 k i=1

P(A) = ∑P(Bi  A) = ∑P(Bi)(A|Bi)

B1

B2

B3

B4

A

Bk

Bn

Partitioning of the sample space S Examples: 1. The population of adults in a small town who have completed the requirements for a college degree were categorize according to sex and employment status as shown in the table. Employed Unemployed Total Male Female 460 140 40 260 500 400

Total 600 300 900 Suppose that 36 of those employed and 12 of those unemployed are members of the Rotary Club. If one of these individuals is selected at random, find the probability than the one selected is a member of the Rotary Club.

1

Roderick B. Astillero

Probability theory

Bayesian Probability

Solution: A: the person selected is a member of the Rotary Club; E: the person is employed; E’: the person is unemployed; E  A: the person is employed and member of the Rotary Club; E’  A: the person is unemployed and member of the Rotary Club.

E A EA E’A

E’

P(A) = P[(EA)  (E’A)] = P(EA) + P(E’A) = P(E)P(A|E) + P(E’)P(A|E’) = (600/900)(36/600) + (300/900)(12/300) = (2/3)(3/50) + (1/3)(1/25) = 4/75 (ans) 2. In a certain assembly plant, three machines, B1, B2, and B3, make 30%, 45%, and 25%, respectively, of the products. It is known from the past experience that 2%, 3%, and 2% of the products made by each machine, respectively, are defective. Suppose that a finished product is randomly selected, what is the probability that it is defective? Solution: A: the product is defective; B1: the product is made by machine B1; B2: the product is made by machine B2; B3: the product is made by machine B3. P(A) = P(B1)P(A|B1) + P(B2)P(A|B2) + P(B3)P(A|B3) = (0.3)(0.02) + (0.45)(0.03) + (0.25)(0.02) = 0.006 + 0.0135 + 0.005 = 0.0245 (ans)

2

Roderick B. Astillero

Probability theory

Bayesian Probability

Exercises 5.1 1. In a certain region of the country it is known from past experience that the probability of selecting an adult over 40 years of age with cancer is 0.05. If the probability of a doctor correctly diagnosing a person with cancer as having the disease is 0.78 and the probability of incorrectly diagnosing a person without cancer as having the disease is 0.06, what is the probability that a person is diagnosed as having cancer? Ans. 0.0960 2. Police plan to enforce speed limits by using radar traps at 4 different locations within the city limits. The radar traps at each of the locations L1, L2, L3, and L4 are operated 40%, 30%, 20%, and 30% of the time, and if the person who is speeding on his way to work has probabilities of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations, what is the probability that he will receive a speeding ticket? Ans. 0.27 3. Suppose that colored balls are distributed in three indistinguishable boxes as follows: Box 1 2 3 Red 2 4 3 White 3 1 4 Blue 5 3 3 A box is selected at random from which a ball is drawn at random. Find the probability that the ball is red. Ans. 1/3 4. A large industrial firm uses 3 local motels to provide overnight accommodations for its clients. From past experience it is known that 20% of the clients are assigned rooms at the Ramada Inn, 50% at the Sheraton, and 30% at the Lakeview Motor Lodge. If the plumbing is faulty in 5% of the rooms at the Ramada Inn, in 4% of the rooms at the Sheraton, and in 8% of the rooms at the Lakeview Motor Lodge, what is the probability that a client will be assigned a room with faulty plumbing? Ans. 0.054 5. Suppose that the four inspectors at a film factory are supposed to stamp the expiration date on each package of the film at the end of the assembly line. John, who stamps 20% of the packages, fails to stamp the expiration date once in every 200 packages; Tom, who stamps 60% of the packages, fails to stamp the expiration date once in every 100 packages; Jeff who stamps 15% of the packages, fails to stamp the expiration date once in every 90 packages; and Pat, who stamps 5% of the packages, fails to stamp the expiration date once in every 200 packages. Find the probability that a customer will complain that her package of film does not show the expiration date. Ans. 0.008917

Examples: 1. In a certain assembly plant, three machines, B1, B2, and B3, make 30%, 45%, and 25%, respectively, of the products. It is known from the past experience that 2%, 3%, and 2% of the products made by each machine, respectively, are defective. If a product is randomly selected and found to be defective, what is the probability that it was made by machine B3? Solution: A: the product is defective; B1: the product is made by machine B1; B2: the product is made by machine B2; B3: the product is made by machine B3. P(B3|A) = P(B3)P(A|B3)/[P(B1)P(A|B1) + P(B2)P(A|B2) + P(B3)P(A|B3)] = (0.25)(0.02)/[(0.30)(0.02) + (0.45)(0.03) + (0.25)(0.02)] = 0.005/(0.06 + 0.0135 + 0.005) = 10/49 (ans) 2. In a certain region of the country it is known from past experience that the probability of selecting an adult over 40 years of age with cancer is 0.05. if the probability of a doctor correctly diagnosing a person with cancer as having the disease is 0.78 and the probability of incorrectly diagnosing a person without cancer as having the disease is 0.06, what is the probability that a person diagnosed as having cancer actually has the disease? Solution: A: the person is diagnosed as having cancer; B: the person has cancer; B’: the person has no cancer. P(B|A) = P(B)P(A|B)/[P(B)P(A|B) + P(B’)P(A|B’)] = (0.05)(0.78)/[(0.05)(0.78) +(0.95)(0.06)] = 0.039/0.096 = 13/32 (ans)

4

Roderick B. Astillero

Probability theory

Bayesian Probability

Exercises 5.2 1. Police plan to enforce speed limits by using radar traps at 4 different locations within the city limits. The radar traps at each of the locations L1, L2, L3, and L4 are operated 40%, 30%, 20%, and 30% of the time. A person who is speeding on his way to work has probabilities of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations. If the person received a speeding ticket on his way to work, what is the probability that he passed through the radar trap located at L2? Ans. 1/9 2. A regional telephone company operates three identical relay stations at different locations. During a one-year period, the number of malfunctions reported by each station and the causes are shown below. Stations: A B C Problems with electricity supplied 2 1 1 Computer malfunction 4 3 3 Malfunctioning electrical equipment 5 4 4 Caused by other human errors 7 7 5 Suppose that a malfunction was reported and it was found to be caused by other human errors. What is the probability that it came from station C? Ans. 45/122 3. A commuter owns 2 cars, 1 a compact and 1 a standard model (a larger one). About ¾ of the time he uses the compact to travel to work, and about ¼ of the time the larger car is used. When he uses the compact car, he usually gets home by 5:30 P.M. about 75% of the time; if he uses the standard-size car, he gets home by 5:30P.M. about 60% of the time. If he gets home at 5:35 P.M., what is the probability that he used the compact car? Ans. 15/23 4. A truth serum has the property that 90% of the guilty suspects are properly judged while, of course, 10% of guilty suspects are improperly found innocent. On the other hand, innocent suspects are misjudged 1% of the time. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent? Ans. 0.174 5. A paint-store chain produces and sells latex and semigloss paint. Based on long-range sales, the probability that a customer will purchase latex is 0.75. Of those that purchase latex paint, 60% also purchase rollers. But 30% of semigloss buyers purchase rollers. A randomly selected buyer purchases a roller and a can of paint. What is the probability that the paint is latex? Ans. 6/7

5

Roderick B. Astillero

Probability theory

Bayesian Probability

Exercises 5.3 1. An allergist claims that 50% of the patients she tests are allergic to some type of weed. What is the probability that (a) Exactly 3 of her next 4 patients are allergic to weeds? (b) None of her next 4 patients are allergic to weeds. 2. From a group of 4 men and 5 women, how many committees of size 3 are possible (a) With no restrictions? (b) With 1 man and 2 women? (c) With 2 men and 1 woman if a certain man must be on the committee? 3. From a box containing 6 black balls and 4 green balls, 3 balls are drawn in succession. What is the probability that all 3 are the same color (a) if each ball is replaced in the box before the next draw is made? (b) if each ball is not replaced before the next draw is made? 4. Pollution of the rivers in the in the United States has been a problem for many years. Consider the following events: A = {The river is polluted.} B = {A sample of water tested detects pollution.} C = {Fishing permitted} Assume: P(A) = 0.3, P(B|A) = 0.75, P(B|A’) = 0.20, P(C|A  B) = 0.20, P(C|A’  B) = 0.15, P(C|A  B’) = 0.80, P(C|A’  B’) = 0.90. (a) Find P(A  B  C). (b) Find P(B’  C). (c) Find P(C). (d) Find the probability that the river is polluted, given that fishing is permitted and the sample tested did not detect pollution. 5. A certain government agency employs three consulting firms (A, B, and C) with probabilities 0.40, 0.35, and 0.25, respectively. From past experience it is known that the probability of cost overruns for the firms are 0.05, 0.03, and 0.15, respectively. (a) What is the probability that a cost overrun will be experienced by the agency? (b) If there is a cost overrun, what is the probability that the consulting firm involved is company C?